Simplicial homotopy method for the solution of nonlinear two‐point boundary value problems

Homotopy methods are useful in determining the solution of nonlinear equations from a remote initial point. Numerous studies have been made to apply the methods to the numerical solution of nonlinear two‐point boundary value problems. However, the method has a disadvantage in that the execution speed is degraded greatly with the increase of the dimension of the system of finite difference equations. This paper proposes an efficient numerical method for the solution of nonlinear two‐point boundary value problems, based on the simplicial homotopy method and the decomposition technique. It is shown first that the system of equations to consider can always be reduced to the system of equations with lower dimension, by applying the decomposition technique to the system of finite difference equations. By this technique, the computational complexity of the algorithm can be decreased drastically compared with the traditional methods. Then an efficient mesh refinement strategy is proposed, taking into consideration the accuracy of the simplicial approximation. Finally, the theorems concerning the convergence of the method are presented and verified by numerical examples. The results are compared with those of the traditional methods, indicating the effectiveness of the proposed method.